For every differentiable function $f(x)$,
is $f(x) = \sum_{n=0} ^ {\infty} \frac {f^{(n)}(0)}{n!} \, (x)^{n}$ always true and can be written?
The only thing we have to be concerned is just whether this taylor series (maclaurin series) converge or not? Therefore, we have to examinine some properties of the derivatives. But all in all $f(x) = \sum_{n=0} ^ {\infty} \frac {f^{(n)}(0)}{n!} \, (x)^{n}$ can always be written?
For example: Does taylor series of $y(t) = t^{3/2}$ exist at $t = 0$?